Problem - Suppose you have a total income of I to spend on three goods x1, x2 and x3, with unit prices p1, p2 and p3 respectively. Your taste can be represented by the utility function u(x1, x2, x3) = x1ax2bx31-a-b where a and b are between 0 and 1, and a + b < 1.
(a) What is your optimal choice for x1, x2 and x3? Use the Lagrange Method.
(b) What are the shares of income spent on the three goods respectively?
(c) Derive your indirect utility function.
(d) Derive your expenditure function.
(e) If there are n goods and the utility function is: u(x1, x2, ... ,xn) = x1a1x2a2 · · · xn1- a1 -a2 - · · · - an-1 where a1, a2,..., an-1 are all between 0 and 1, and a1+a2+...+an-1 < 1.
The unit prices are p1 , p2,..., pn respectively.
Without calculations, write down the demand function for x1 and the share of income spent on it.